Characteristically simple and non-abelian implies monolith in automorphism group

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Statement

If G is a Characteristically simple group (?) that is non-abelian, then G is a Monolith (?) in its Automorphism group (?). In particular, the automorphism group of G is a Monolithic group (?).

In particular, this shows that the automorphism group of any Simple non-abelian group (?) is a complete group.

Related facts

Analogue for abelian groups and more general groups

Other related facts

Facts used

  1. Center is characteristic
  2. Centerless implies NSCFN in automorphism group
  3. Characteristically simple and NSCFN implies monolith: If G is a characteristically simple subgroup of a group H, such that G is normal in H, C_H(G) \le G, and every automorphism of G extends to an inner automorphism of H, G is a the unique nontrivial normal subgroup of H contained in every nontrivial normal subgroup of H.

Proof

Given: A characteristically simple non-abelian group G, with automorphism group \operatorname{Aut}(G).

To prove: \operatorname{Aut}(G) is complete: it is centerless and every automorphism of it is inner.

Proof:

  1. (Given data used: G is characteristically simple non-abelian; Facts used: fact (1)): G is centerless: By fact (1), the center of G is characteristic in G. Since G is non-abelian, the center of G is not equal to G. Since G is characteristically simple, this forces the center of G to be trivial.
  2. The natural mapping from G to \operatorname{Aut}(G) given by the conjugation action is injective, identifying G with its image \operatorname{Inn}(G): This follows from the fact that G is centerless, so the kernel of the map is trivial.
  3. (Facts used: fact (2)): G is normal, self-centralizing and fully normalized in \operatorname{Aut}(G): This follows from fact (2), and the fact that G is centerless.
  4. (Given data used: G is characteristically simple; Facts used: fact (3)): By fact (3) and the conclusion of the previous step (step (3)), and the given datum that G is characteristically simple, we conclude that G is a monolith in \operatorname{Aut}(G): it is the unique nontrivial normal subgroup of \operatorname{Aut}(G) that is contained in every nontrivial normal subgroup of \operatorname{Aut}(G).